Optical Fiber Pressure Sensor with Self-Temperature Compensation Structure Based on MEMS for High Temperature and High Pressure Environment

Abstract

To meet the pressure measurement requirements of deep earth exploration, we propose an OFPS (optical fiber pressure sensor) with self-temperature compensation based on MEMS technology. A spectral extraction and filtering algorithm, based on FFT (fast Fourier transform), was designed to independently demodulate the composite spectra of multiple FP (Fabry–Pérot) cavities, enabling the simultaneous measurement of pressure and temperature parameters. The sensor was fabricated by etching on an SOI (silicon on insulator) and bonding with glass to form pressure-sensitive FP cavities, with the glass itself serving as the temperature-sensitive component as well as providing temperature compensation for pressure sensing. Experimental results showed that within the pressure range of 0–100 MPa, the sensor exhibited a sensitivity of 0.566 nm/MPa, with a full-scale error of 0.34%, and a linear fitting coefficient (R²) greater than 0.9999. Within the temperature range of 0–160 °C, the temperature sensitivity of the glass cavity is 0.0139 nm/°C and R² greater than 0.999.

1. Introduction

Deep earth exploration and deep resource development are cutting-edge fields in current geological exploration and engineering technology. According to the IHTHPA (International High Temperature and High Pressure Association), wells are classified as high temperature and high pressure if the formation temperature exceeds 300° F (149 °C), the formation pressure exceeds 15,000 psi (103.4 MPa), or the wellhead pressure exceeds 10,000 psi (68.9 MPa) [1]. Accurate pressure measurement is essential to ensure the safety of resource extraction, monitor the subsurface fluid dynamics, and maintain equipment stability. Traditional electronic pressure sensors are widely used in industrial applications due to their established technology and low cost [2,3,4]. In recent years, OFSs have experienced rapid development and demonstrated exceptional performance in harsh environments including high temperatures, high pressures, and strong electromagnetic interference. Particularly in applications requiring power-free operation and long-distance transmission, OFSs offer distinct advantages including ease of integration and enhanced resistance to interference [5,6,7].

Currently, there are two main approaches for OFPSs used in high-pressure measurements: fiber Bragg grating (FBG) and FP interference. However, the pressure sensitivity of bare gratings is relatively low, requiring the use of complex sensitization structures that can compromise the accuracy and stability of the sensor [8,9,10]. Pressure sensors based on the FP interference principle offer advantages such as a simple structure, high sensitivity, and high reliability, making them suitable for pressure measurements in extreme environments. The traditional method for fabricating fiber optic FP cavities typically involves etching the fiber tip and then depositing various materials, such as silicon [11], silicon dioxide [12], polymers [13], and others, to create sensitive diaphragms that make the FP cavity responsive to pressure. These studies are generally conducted within a small pressure range (0–0.1 MPa) for biomedical pressure monitoring. Some progress has been made in the research of FP pressure sensors for high-pressure environments, for example, Gao et al. [14] proposed a four-segment structure consisting of two single-mode fibers (SMFs), one hollow core fiber (HCF), and one coreless fiber (CF), which were fusion spliced to enable pressure measurement in the range of 0–10 MPa. To enhance pressure resistance performance, J. Ma et al. [15] fabricated a pressure-sensitive microbubble FP cavity on the fiber end face using arc welding, achieving pressure measurements up to 40 MPa, and Zhao et al. [16] utilized a quartz capillary and optical fiber to form a FP cavity, performing pressure sensing by altering the cavity length through lateral compression of the capillary. The highest measured pressure was 69 MPa. However, the aforementioned FP cavity is challenging to produce at scale, which hinders the ability to achieve consistent sensing demodulation. Additionally, the voltage resistance range remains inadequate and requires further improvement.

MEMS technology primarily focuses on microelectronics and micro-mechanical processing. When applied to the production of sensor chips for diaphragm-type fiber FP pressure sensors, it offers several advantages including high dimensional accuracy, the potential for mass production, low cost, flexible design options, and excellent consistency [17]. These benefits collectively contribute to the enhanced stability of the sensor. Currently, most studies on fabricating fiber optic FP pressure sensors based on the MEMS process use silicon as the sensitive diaphragm to form the FP reflector. These studies predominantly focused on low-pressure ranges [18,19,20,21]. For high-pressure detection, Fu et al. [22] employed monocrystalline silicon etching and glass bonding to fabricate pressure sensors, achieving a maximum pressure measurement of 30 MPa, while Qi et al. [23] proposed an embedded structure designed to withstand high pressure, which involves etching cavities into glass wafer, bonding it with single-crystal silicon to create sensing diaphragms, and embedding the diaphragm into the glass tube. This configuration enables pressure measurements in the range of 2–120 MPa. However, the isotropic nature of glass leads to lateral corrosion during the etching process, which adversely affects the stability and accuracy of the sensor structure.

Sensors based on the MEMS process typically consist of multiple FP cavities. To obtain independent spectra, it is necessary to decompose the composite spectrum, and the commonly used demodulation algorithms include the cross-correlation algorithm and the Fourier transform. The cross-correlation algorithm determines the actual cavity length by correlating the sensor spectrum with an ideal reference spectrum and identifying the maximum correlation [24,25]. Fourier transform is the process of converting signals from the wavelength domain to the frequency domain. Since the spectrum of a multi-cavity FP sensor consists of signals at different frequencies, this method can be used to separate the individual spectral components. Gao et al. [14] used the fast Fourier transform (FFT) method to implement dual-parameter sensing for temperature and pressure. Liu et al. [26] employed the FFT and Fourier band pass filtering (FBPF) methods to analyze the temperature and pressure sensitivities of two different cavities. Additionally, Guo et al. [27] applied the FFT with the minimum mean square error (FFT-MMSE) algorithm to demodulate the optical cavity length of the sensor, enabling temperature and pressure measurements in the ranges of 25 °C to 600 °C and 0 to 10 MPa.

In this paper, an FP-OFPS based on MEMS technology is proposed. The cavity was fabricated on an SOI using ICP etching technology and bonded with BF33 glass with a low thermal expansion coefficient to form a FP cavity, which can measure pressure. In addition, the glass itself forms a FP cavity, which functions as a temperature sensor, enabling self-temperature compensation of the structure. A testing platform was developed to perform performance testing and analysis of the sensor across a pressure range of 0–100 MPa and a temperature range of 0–160 °C. The sensor exhibited a pressure sensitivity of 0.56 nm/MPa, with an accuracy of 0.34% F.S. The temperature compensation structure demonstrated a sensitivity of 0.0139 nm/°C, and the nonlinear fitting of the pressure sensing structure to temperature exceeded 0.999. Figure 1 systematically illustrates the technical roadmap of this study.

2. Methods

2.1. Fabry–Pérot Cavity Sensing Principle

An FP cavity consists of two parallel mirrors, one typically fixed and the other movable. When light passes through the FP cavity, the distance L between the mirrors creates a difference in the optical path length, which leads to interference. According to the principle of multi-beam interference, the reflected light intensity is given by [28],

$$I_R={\displaystyle \frac{2R\left(1-\cos \frac{4\pi }{\lambda }{n}_{0}L\right)}{1+R^2-2R\cos \frac{4\pi }{\lambda }{n}_{0}L}}{I}_0,$$

(1)

where R is the radius of the FP cavity, $n_0$ is the refractive index of the medium inside the cavity, $\lambda$ is the wavelength of the incident light, and L is the cavity length. In the interference pattern of the FP cavity, the fringe reaches an extreme when the phase difference $\delta$ satisfies:

$$\begin{array}{l}\delta ={\displaystyle \frac{2\pi \Delta }{{\lambda }_m}}=2m\pi m=0, 1, 2\dots \dots \\ \Delta =2n_{0}L ,\end{array}$$

(2)

where $\Delta$ is the optical path difference. Therefore, for a particular peak of the fringe, it satisfies:

$$L=\mathit{const}\cdot {\lambda }_m ,$$

(3)

When the cavity length changes, the peak resonance condition shifts, leading to the expression:

$${\displaystyle \frac{\Delta L}{L}}={\displaystyle \frac{\Delta \lambda }{\lambda }},$$

(4)

When the structure of the FP cavity is fixed, meaning that the values of R and $n_0$ remain constant, the reflected light intensity becomes a function of the wavelength λ and the cavity length L. When external environmental parameters such as pressure and temperature change, the cavity length of the FP cavity changes. The change in cavity length can be obtained by demodulating the wavelength shift, thus realizing the measurement of pressure parameters.

2.2. Diaphragm Deformation Principle

The core sensing element of the MEMS fiber-optic pressure sensor is the SOI diaphragm, which deforms when external pressure is applied to its surface. According to thin-plate bending theory, the deformation of the diaphragm under a uniform pressure P can be calculated using the following equation [29]:

$$\omega (r)={\displaystyle \frac{3(1-{\nu }^2){(r^2-a^2)}^2}{16E{T}^3}},$$

(5)

where E is the Young’s modulus of the silicon diaphragm, T is the diaphragm’s thickness, and ν is the Poisson’s ratio of silicon. At r = a, where r represents the radial distance from the center and a is the radius of the diaphragm, the deformation is maximized at the center, with the deformation gradually decreasing toward the edges. The diaphragm’s deformation is influenced by factors such as the applied pressure, the diaphragm’s geometry, and the material’s elastic modulus. According to the theory of small deflections for thin plates, the maximum deformation should be less than one-fifth of the diaphragm’s thickness to maintain linear sensor behavior.

$${\omega }_{\max }={\displaystyle \frac{3P{R}^4\left(1-v^2\right)}{16E{T}^3}}<{\displaystyle \frac{T}{5}},$$

(6)

By satisfying this condition, the diaphragm deformation remains within the small deformation range, ensuring that the sensor output retains good linearity. Meanwhile, the sensitivity of the diaphragm is given by the following equation [29]:

$$S={\displaystyle \frac{d{\omega }_{\max }}{dP}}={\displaystyle \frac{3R^4\left(1-v^2\right)}{16E{T}^3}},$$

(7)

From this formula, it is evident that increasing the effective radius of the diaphragm and reducing its thickness will result in higher sensor sensitivity.

2.3. Mechanical Simulation

Finite element mechanical simulations of the SOI–glass composite diaphragm were conducted to assess its structural stability and overload resistance. The constitutive model included SOI layers (comprising substrate silicon, silicon dioxide, and top silicon) and glass layers, with a cavity formed in the SOI to serve as the FP cavity, as shown in Figure 2a. The geometric parameters of the model were input into the simulation software, and the physical properties of the materials were defined as outlined in

Table 1. Physical properties of the materials.

Material	Young’s Modulus (GPa)	Poisson’s Ratio	CTE * (/°C)
Si	130	0.28	2.6 × 10−6
SiO2	70	0.17	0.5 × 10−6
BF33 Glass	64	0.21	3.25 × 10−6

* CTE—coefficient of thermal expansion.

.

The boundary conditions were set to fully fix the bottom and perimeter of the structure, simulating the support conditions encountered in practical use. A steady-state pressure of 10 MPa was applied to the outer surface of the SOI as the external load. The three-dimensional deformation distribution of the diaphragm, as shown in the simulation results in Figure 2b, revealed that under uniform pressure, the maximum deformation occurred at the center of the diaphragm, and the deformation gradually decreased as it approaches the edges. This observation is consistent with the theoretical analysis presented in Equation (5). In addition, the effects of varying diaphragm thicknesses and FP cavity radii on diaphragm deformation were analyzed, as shown in Figure 2c. The deformation at the center of the diaphragm decreased with an increase in diaphragm thickness and a decrease in the FP cavity radius.

The final design structural parameters are presented in

Table 2. Parameters of the structural models.

Parameters	Value (μm)	Description
r	900	FP cavity radius
l	100	FP cavity length
a	5000	Diaphragm side length
H	650	Glass thickness
T	350	Substrate silicon thickness

. A pressure load ranging from 0 to 100 MPa, applied in increments of 10 MPa, was used to obtain the linear fitting plots of the diaphragm center deformation under varying pressure loads, as shown in Figure 2d. The diaphragm deformation increased linearly with the applied load, and the cavity length sensitivity was calculated to be 0.0386 μm/MPa, corresponding to a wavelength sensitivity of 0.5983 nm/MPa (assuming the center wavelength of the incident light is 1550 nm). Furthermore, the maximum stress in the structure increased gradually with the external pressure in the range of 0 to 100 MPa. As shown in Figure 2e, the maximum stress in the structure, which occurred at the edge of the FP cavity under a 100 MPa load, was approximately 1.026 GPa, remaining below the maximum allowable stress for silicon [23]. Therefore, based on the simulation analysis, the designed sensing diaphragm is structurally stable and can maintain its integrity under a 100 MPa pressure condition.

2.4. Spectra Simulation and Demodulation Algorithm

Based on the determination of the structural parameters of the model, spectral simulations of the sensing diaphragm were conducted by inputting the refractive index of the material and the thickness parameters. Due to the presence of multiple reflective surfaces within the structure, the light beam undergoes multiple reflections and interference within the sensing diaphragm. As a result, the final collected signal is not an ideal sinusoidal waveform, but rather a spectral signal formed by the interference from both the air cavity (FP_cav) and the glass cavity (FP_glass), as shown in Figure 3.

When external pressure is applied to the diaphragm surface, the silicon diaphragm deforms and bends, resulting in a change in the cavity length of FP_cav, while FP_glass remains fixed and is almost unaffected by the pressure. Additionally, temperature variations cause the glass, silicon, silica, and the air inside the cavity to expand or contract, leading to changes in the cavity lengths of both FP_cav and FP_glass. Therefore, the sensitivity of FP_cav to temperature variations can impact the accuracy of its pressure measurements. The temperature and pressure cross-sensitivity of FP_cav can be compensated by the temperature measurement of FP_glass, and the temperature and pressure variations can be obtained from the following matrix,

$$\left[\begin{array}{c}\Delta L_{glass}\\ \Delta L_{cav}\end{array}\right]=\left[\begin{array}{cc}{S}_{glass-T}& 0\\ S_{cav-T}& -S_P\end{array}\right]\left[\begin{array}{c}\Delta T\\ \Delta P\end{array}\right],$$

(8)

where $\Delta L_{cav}$ and $\Delta L_{glass}$ represent the variations in the cavity lengths of FP_cav and FP_glass, respectively, and $\Delta T$ and $\Delta P$ denote the variations in temperature and pressure, respectively. By performing a matrix inversion, the variations in temperature and pressure can be determined using the following matrix:

$$\left[\begin{array}{c}\Delta T\\ \Delta P\end{array}\right]={\displaystyle \frac{1}{S_{glass-T}{S}_P}}\left[\begin{array}{cc}{S}_P& 0\\ S_{cav-T}& -S_{glass-T}\end{array}\right]\left[\begin{array}{c}\Delta L_{glass}\\ \Delta L_{cav}\end{array}\right],$$

(9)

In this study, a spectrum extraction and filtering algorithm based on Fourier transform was employed to separate the spectra of the two FP cavities. First, the spectral data were uniformly converted to the wavenumber domain using cubic spline interpolation, which eliminates errors introduced by uneven wavelength spacing. The mathematical expression for the interpolation is as follows:

$$S_{\mathit{int}erp}(k)={\displaystyle \sum _{i=1}^n{S}_i}\cdot {\varphi }_i(k),$$

(10)

where $S_{interp}\left(k\right)$ represents the spectral intensity after interpolation at the wavenumber, $S_i$ is the spectral intensity of the original data point, and ${\varphi }_i\left(k\right)$ is the interpolation basis function. The interpolated spectra were then windowed using the Hanning window function to reduce spectrum leakage and side-lobe effects that occur during Fourier transform. Fast Fourier transform (FFT) was applied to the windowed spectral data to convert it from the time domain to the frequency domain, as expressed in the following equation:

$$S_{\mathrm{FFT}}(f)={\displaystyle \sum _{k=0}^{N-1}{S}_{windowed}}(k)e^{-i2\pi fk/N},$$

(11)

where $S_{windowed}\left(k\right)$ denotes the windowed function, $S_{\mathrm{F}\mathrm{F}\mathrm{T}}\left(f\right)$ represents the FFT result at the frequency, N is the number of sampling points of the signal, and k is the index in the time or wavelength domain. The FFT frequency spectrum revealed the distribution of various frequency components in the spectral signal. To extract the low-frequency and high-frequency components, the algorithm employs a bandpass filter to process the frequency-domain signal. The transfer function of the bandpass filter is given by:

$$H(f)=\left\{\begin{array}{c}1\\ 0\end{array}\right. \begin{array}{c}if f_{low}\le \left|f\right|\le f_{high}\\ otherwise\end{array},$$

(12)

where $f_{low}$ and $f_{high}$ represent the cutoff frequencies for the low- and high-frequency components, respectively. The inverse Fourier transform (IFFT) was then applied to the filtered frequency-domain signal, converting it from the frequency domain back to spectra in the wavelength domain,

$$S_{\mathrm{IFFT}}(k)={\displaystyle \sum _{f=0}^{N-1}{S}_{filtered}}(f)e^{i2\pi fk/N},$$

(13)

After applying this algorithm to process the simulated spectrum in Figure 3, as shown in Figure 4a, the FFT spectrum displayed three different frequency components corresponding to FP_cav, FP_glass, and FP_com, respectively, formed by both, sequentially from low frequency to high frequency. Illustrations (i) and (ii) show the independent interference spectrum corresponding to FP_cav and FP_glass after bandpass filtering, respectively.

In the simulation, the cavity lengths of both FP_cav and FP_glass were set to increase by 0.3 μm with a step size of 0.1 μm, and the FFT spectra processed by the algorithm are shown in Figure 4b. As the cavity length increased, the free spectral range (FSR) of the FP cavity decreased, leading to a reduction in the period and an increase in frequency. Consequently, all three frequency peaks shifted to the right.

3. Fabrication

This study employed MEMS technology to fabricate the pressure sensors, prepare the FP cavities on SOI wafer using etching techniques, and bond the SOI to the glass wafer to form sensing diaphragms. Finally, the diaphragm was adhesively fixed to the fiber collimator, and the entire assembly was sealed within the package to form a complete sensor.

3.1. Sensor Diaphragm Fabrication

A 4-inch SOI wafer (with a substrate silicon thickness of approximately 380 μm and a thickness of the top layer silicon and silicon dioxide of about 100 μm in total) was selected as the sensitive diaphragm for the sensor. The processing of SOI wafers primarily involves the following steps, as shown in Figure 5a–f: (a) Cleaning: the surface of the SOI wafers is thoroughly cleaned using RCA cleaning technology to remove impurities and contaminants; (b) Spin coating: a uniform layer of photoresist is applied to the wafer; (c) Patterning: after exposure and development, transfer the pattern on the mask to the SOI wafer; (d) Etching: ICP etching technology is used to etch the top silicon layer, and the SiO₂ layer is removed through wet etching, thereby forming a circular cavity on the SOI; (e) Deposition: SiO₂-Ta₂O₅ optical interference film is prepared by HiPIMS (high power impulse magnetron sputtering) reactive sputtering to enhance the reflection performance of the FP cavity; and (f) Photoresist removal: an acetone solution is used to remove the photoresist, followed by cleaning and drying to obtain the processed SOI wafers.

Borosilicate glass (BF33) wafers with a low coefficient of thermal expansion as well as high transparency and stability were selected as the substrate and optical transmission material for the sensing diaphragm. The processing of glass wafers included the following steps, as shown in Figure 5g–l: (g) cleaning; (h) spin coating photoresist; (i) patterning; (j) deposition, where the size of the coating on the glass must be slightly smaller than the SOI etching cavity to ensure proper bonding between the two; (k) photoresist removal; and (l) wafer bonding, where the processed silicon and glass wafers are bonded using anodic bonding under high temperature and pressure conditions, with a temperature range of 350–400 °C and a pressure of approximately 1–3 MPa. An annealing treatment was then applied to the bonded wafer interface to remove residual stress and enhance the interface quality.

Figure 6a,b shows the bonded wafer and the single sensing film obtained after slicing, respectively. The cross-sectional morphology of the diaphragm, observed using SEM (Scanning Electron Microscopy, Carl Zeiss AG, Oberkochen, Germany), are shown in Figure 6c,d. The SEM image revealed a flat surface of the sensing film and the interface between the SOI and glass bonding, with no apparent defects. Additionally, the deposited film layer appeared dense, with a clearly defined separation between layers.

3.2. Sensor Packaging

To ensure efficient optical signal transmission, it is essential that the incident light enters the FP cavity vertically. We completed the alignment operation on the specialized optical fiber-processing platform (3SAE Large Diameter Splicer, 3SAE Technologies Inc., Franklin, TN, USA, LDS 2.5), secured the diaphragm on the micro displacement stage using vacuum adsorption, and fixed the fiber collimator at the opposite end. Through the built-in program, fine adjustments could be made in ten dimensions to ensure precise alignment between the fiber collimator and the center of the sensing diaphragm as well as vertical incidence of light. Meanwhile, the demodulator was used to monitor the spectral signal. The precision alignment platform constructed is shown in Figure 7a,b. After alignment, a UV adhesive was applied and cured at the interface between the diaphragm and the fiber collimator to finalize the fixation, resulting in the structure shown in Figure 7c.

The package structure was designed using 316-steel for pressure testing of the diaphragm. During the encapsulation process, a small amount of optical fiber adhesive 353ND (EPO-TEK, Billerica, MA, USA) was evenly applied around the bottom of the reserved grooves in the structure to prevent glue from overflowing and contaminating the center of the diaphragm after placing it. After the glue solidified, a layer of vulcanized rubber was applied to the outer surface of the diaphragm to seal it, ensuring that there was no leakage during the testing process, as shown in Figure 8c,d.

4. Setup and Experimental Results

4.1. Setup

The pressure testing platform was constructed as shown in Figure 9a, consisting of the following components: a desktop high-pressure pump (pressure range 0–280 MPa, accuracy 0.1 Pa), a digital pressure gauge (pressure range 0–250 MPa, accuracy 0.05), an MOI fiber grating demodulator (wavelength range 1500–1600 nm, wavelength accuracy 1 pm), and the sealed sensor, which was fixed to the M20 × 1.5 fast interface of the high-pressure pump via an adapter.

The temperature testing platform was constructed as shown in Figure 9b, consisting of the following components: an intelligent precision temperature block (temperature range −40–160 °C, accuracy ±1 °C), an intelligent standard thermometer (accuracy ±0.025 °C @ 160 °C), and an MOI fiber grating demodulator (wavelength range 1500–1600 nm, wavelength accuracy 1 pm).

4.2. Pressure Test

The pressure test was performed at room temperature (25 °C), and the pressure test range was increased from 0 MPa to 100 MPa and then decreased back to 0 MPa, with a step size of 10 MPa. After each increase, the pressure was gradually raised and maintained for 10 min to ensure spectral stability before collecting the spectral data. The raw spectra were then processed using the designed filtering algorithm. FP_glass is insensitive to pressure, so only the FP_cav needs to be demodulated. The interference spectra are shown in Figure 10a. As the pressure increased, the sensor diaphragm deformed, leading to a reduction in the cavity length of the FP cavity and a blue shift in the spectrum. The pressure sensitivity of the sensor was fitted as shown in Figure 10b. The sensitivity during the pressure rise process was 0.56044 nm/MPa, with a linear fitting degree of R² of 0.99995. The sensitivity during the pressure drop process was 0.56089 nm/MPa, with an R² of 0.99996, indicating high linearity. Moreover, fitting both together resulted in a sensitivity of 0.56067 nm/MPa and an R² of 0.99996, which is consistent with the simulation data. In the experiment, we observed that when the pressure reached 30 MPa, the spectral shift had already crossed one FSR. Therefore, we supplemented the pressure rise experiment with a step size of 5 MPa from 0 to 25 MPa. The fitting of the spectral shift and sensitivity is shown in Figure 10c,d, with a linearity of 0.99998 and a sensitivity of 0.56006 nm/MPa, which is consistent with the pressure range of 0 to 100 MPa.

When the spectral shift exceeds one FSR range, it may cause difficulty in distinguishing the interference order of spectral fringes. To address this, we used the FFT spectrum to obtain the rough optical path difference [30] and calculate the interference order m corresponding to the valley wavelength ${\lambda }_m$ in the independent FP cavity interference spectrum using Equation (2). At this point, changes in the external environmental parameters can be demodulated by tracking the movement of the valleys at specific interference orders. Taking 0 and 30 MPa as examples, at 0 MPa, the optical path difference ($\Delta$) obtained from the FFT spectrum was approximately 201 μm, resulting in an interference order (m) of 129 at a wavelength of 1563.2 nm. At 30 MPa, $\Delta$ was about 199 μm, with an interference order of 128 at 1558.7 nm and 129 at 1546.8 nm. This approach resolves the issue of interference order ambiguity and enables demodulation over a wide dynamic range.

Meanwhile, the error of the sensor was analyzed, with the nonlinear error defined as the deviation between the actual measured value and the ideal value, and the hysteresis error was defined as the degree to which the output–input curves of the sensor did not coincide during positive (increasing input) and negative (decreasing input) strokes. As illustrated in Figure 11, the sensor reached a maximum nonlinear error of about 0.34% at 0 MPa during the step-down process, while the maximum hysteresis error was about 0.149% at 10 MPa.

4.3. Temperature Test

The temperature test was performed at normal pressure, with a temperature range of 0 °C to 160 °C and a step size of 10 °C. After the spectrum stabilized, we recorded the data and used the designed filtering algorithm to process the original spectrum. Since both FP cavities were sensitive to temperature, their sensitivities were calculated separately. The experimental results showed that as the temperature increased, the spectral wavelengths of both FP_glass and FP_cav underwent a red shift, as shown in Figure 12a,b, separately, which was consistent with the theoretical expectations. The temperature sensitivity fitting curves of FP_glass are shown in Figure 12c. FP_glass exhibited a temperature sensitivity of 0.0139 nm/°C and a linearity of R² = 0.99905.

The variation of the FP_cav cavity length with temperature was nonlinear, as shown in Figure 12d, and in order to verify its repeatability, we conducted three heating and cooling experiments in the temperature range of 0–160 °C under identical experimental conditions. Fitting the experimental data, it was found that the relationship between the sensor output and temperature could be described by an exponential model. The coefficient of determination R² of the fitting result was 0.999. This nonlinearity may have originated from the different thermal expansion coefficients of the glass and silicon materials, resulting in thermal stress, residual air and residual stress generated during the bonding process, etc. These factors may lead to complex changes in the shape and size of the membrane, which often do not follow a simple linear relationship, resulting in a nonlinear response of the cavity length.

For a given temperature point, the temperature sensitivity of FP_cav is the slope of the tangent line at that point, which corresponds to the value of the derivative at that point:

$$S_T(T_0)={\left.{\displaystyle \frac{dy}{dT}}\right|}_{T=T_0},$$

(14)

therefore, by improving Equation (9), the sensitivity matrix obtained for the sensor is as follows:

$$\left[\begin{array}{c}\Delta T\\ \Delta P\end{array}\right]={\displaystyle \frac{1}{210.64}}\left[\begin{array}{cc}36.13& 0\\ S_{cav-T}\left(T_0\right)& -5.83\end{array}\right]\left[\begin{array}{c}\Delta L_{glass}\\ \Delta L_{cav}\end{array}\right],$$

(15)

By measuring the temperature coefficient of the FP_glass cavity, self-temperature compensation of the structure was achieved, thereby reducing the temperature–pressure cross-sensitivity of FP_cav.

5. Discussion

This article presented a FP-OFPS based on multi-beam interference, capable of accurate pressure measurements in high-pressure environments ranging from 0 to 100 MPa, with self-temperature compensation across a temperature range of 0 to 160 °C. While most existing OFPSs rely on external temperature sensors, such as FBGs [31,32,33], for temperature compensation, this study utilized the FP_glass within the structure for independent temperature measurement, enabling self-compensation. This approach reduces the system complexity, enhances the system stability and sensor accuracy, and is better suited for miniaturization and integrated design.

Table 3. Performance of some representative FP pressure sensors.

Scheme	Sensitivity (nm/MPa)	Range (MPa)	Sensitivity (pm/°C)	Range (°C)	Ref.
Microbubble cavity	0.315	0–40	1.55	0–600	[15]
Capillary splicing	216	0–69	/	/	[16]
MEMS silicon-glass	46.94	0–30	/	/	[22]
MEMS embedded structure	1.071 rad/MPa	2–120	2.665 × 10−3 rad/°C	0–60	[23]
This work	36.13	0–100	13.9 (temperature compensation)	0–160

presents a concise comparison between the proposed sensor and other FP schemes reported in the recent literature.

The hysteresis phenomenon observed in pressure measurements has been reported in other MEMS OFPS studies [23], often linked to the material properties of the diaphragm and its mechanical behavior. To mitigate the hysteresis effect caused by material or structural factors, pressure cycling tests are commonly employed [34]. In these tests, the pressure is varied repeatedly over multiple cycles, allowing the sensor to experience sufficient loads within a specific range, thereby reducing measurement errors and drift.

This study demonstrated through theoretical analysis that temperature compensation for pressure sensing can be achieved via the temperature sensitivity of the glass cavity. However, due to limitations in the experimental conditions, pressure testing was not carried out under variable temperature conditions. In future research, we plan to design pressure control equipment capable of withstanding high temperatures to conduct experiments under both variable temperature and pressure conditions, thereby validating the effectiveness of self-temperature compensation. Additionally, further research can focus on integrating high-temperature-resistant materials, improving the temperature compensation mechanisms, and optimizing sensor structures to expand their operating temperature and pressure ranges, thereby enhancing sensor performance in more challenging environments.

6. Conclusions

In summary, we proposed an OFPS based on MEMS technology, utilizing a composite structure of SOI and glass along with the principle of FP cavity interference for accurate high-pressure measurements. The sensor employed FFT-based spectral extraction and filtering algorithms to process the spectra from composite FP cavities, enabling independent demodulation of the two cavities. Changes in external pressure and temperature can lead to variations in the length of the air FP cavity. However, the FP cavity formed by glass is solely sensitive to temperature, and can act as a temperature compensation mechanism to mitigate the cross-sensitivity effect between the air FP cavity’s temperature and pressure. The experimental results demonstrated that the sensor exhibited a good linear response within the pressure range of 0–100 MPa, with a cavity length sensitivity of 0.56 nm/MPa, a nonlinear error of 0.34% F.S, and a hysteresis error of 0.149% F.S. In the temperature range of 0–160 °C, the glass cavity showed a temperature sensitivity of 0.0139 nm/°C, and the nonlinear fitting correlation coefficient of the air FP cavity to temperature was greater than 0.999, indicating that the model can accurately describe the variation in sensor output with temperature. This sensor offers advantages such as compact size, high accuracy, and excellent consistency, making it suitable for wide-ranging applications in pressure and temperature measurements, particularly in fields such as deep earth exploration and resource development.